There is a "real risk" that patient data will be released tomorrow, Simon Harris has said.
The Minister for Further and Higher Education said "there's some evidence that it may already have happened in some instances and that's been verified by the gardaí".
Chief executive of the HSE, Paul Reid, has admitted the process of dealing with the impact of the cyberattack on the HSE could take weeks rather than days.
However, Mr Reid said a decryption key, given to Irish authorities by the hackers on Thursday, helps the process.
“Ultimately, we are seeing a process of weeks rather than days of impact. We have made some good steady progress throughout the weekend. Certainly, the decryption key and script does help us in the process," he said.
He said that over the next few weeks, some hospitals will regain access to national systems, including patient administration systems, but that it is a “slower process".
Hackers targetted the HSE last week with ransomware, causing a complete shutdown of its systems.
Some personalised medical data of Irish patients has been shared online in a bid by the attackers, a Russian group known as Wizard Spider, to further its claims for a $20m ransom, which the State has so far insisted it will not pay.
There was also a thwarted attack on the Department of Health's website on Thursday when the National Cyber Security Centre (NCSC) was contacted by department staff.
Simon Harris said that it's "vital", people know that the state "will never call asking for your bank details over the telephone or by email".
The advice from the government is to immediately contact gardaí.
There have been daily meetings with the departments of Justice, Health and Communications. However, Mr Harris added there is "absolutely no evidence at all" that the systems being used by the HSE are the reason for the cyber attack.
"This is not something that's in any way unique to Ireland, and indeed our cybersecurity experts actually managed to successfully thwart an attack on the Department of Health," he said.
"Of course there'll be lessons to be learned, but the programme of investment here has been very, very significant, and the enemy here is the criminal gangs who are causing absolute chaos.
"If you actually look at the scale of increased staffing and the budget that's gone into the National Cybersecurity Centre in recent years it has been increasing year on year."
Mr Harris added there have "been some encouraging developments" in relation to getting health systems back up and running.
A statement from government released on Sunday afternoon read: "The Government is aware of the risk that data stolen from the HSE may be abused by criminals and is taking measures to prevent that and to support anyone affected.
"There is, sadly, is a real risk of patients’ data being abused in this way. The Government is grateful to mainstream media and online services for their support and we appeal to anyone who may come across this data online not to share it but instead to report it using the tools provided by platforms.
"People are also advised to be cautious of criminals taking advantage of fears around the HSE attack by contacting them to attempt to obtain information or payments. Any such attempts should be reported to An Garda Síochána."